Find the general solution of the differential equation

Question

find the general solution of the given differential equation

(x-2)y'+y=x^(2)-19

(use C as an arbitrary constant)

y=

Answer 1

Rewrite

y'+y/(x-2)= (x²-19)/(x-2) = y'(x)+p(x)y=q(x)

Find integration factor:

IF = e^∫dx/(x-2)=e^ln(x-2)=x-2

Multiply thru by the IF

(x-2)y' +y = x²-19

((x-2)y)' = x²-19

Integrate both sides

(x-2)y = x³/3-19x+C

y = (x³/3-19x+C)/(x-2)